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To determine whether 179 is a prime number, let's follow the instructions in the image.

### Step-by-Step Solution:

1. **Approximate Square Root Calculation:**
   - Given that \( w \approx 13.4 \) is an approximation to \( \sqrt{179} \).
   - To determine the primality of 179, we only need to check divisibility by prime numbers less than or equal to \( w \).

2. **Prime Divisors to Check (below 13.4):**
   - The relevant prime numbers less than or equal to 13.4 are: 2, 3, 5, 7, 11, and 13.

3. **Checking Divisibility:**
   - For each number in the list (2, 3, 5, 7, 11, and 13), we determine if it is a divisor of 179:
     - **2**: 179 is odd, so it is not divisible by 2.
     - **3**: Sum of digits \( 1 + 7 + 9 = 17 \), which is not divisible by 3.
     - **5**: 179 does not end in 0 or 5, so it is not divisible by 5.
     - **7**: \( 179 \div 7 \approx 25.57 \), not an integer, so it’s not divisible by 7.
     - **11**: Alternate sum rule \( (1 + 9) - 7 = 3 \), which is not divisible by 11.
     - **13**: \( 179 \div 13 \approx 13.77 \), not an integer, so it’s not divisible by 13.

4. **Conclusion:**
   - Since 179 is not divisible by any prime number less than or equal to \( \sqrt{179} \), it is a **prime number**.

### Selecting the Options:

- **For 2, 3, 5, 7, 11, and 13**: Choose "less than w" for each since these primes are below 13.4.
- **For other primes (17, 19, 23, 29, 31)**: Choose "greater than w" as they are above 13.4 and do not need to be checked.
  
Finally, select "prime" for 179, as it is confirmed to be a prime number.

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### Related Questions

1. How do we find the square root approximation for large numbers when determining primality?
2. Why is it only necessary to check up to the square root when testing for primality?
3. How does the divisibility test work for different primes, such as 3, 7, or 11?
4. Could a composite number have no factors below its square root?
5. How can we check primality for very large numbers more efficiently?

### Tip:
When testing for primality, checking divisibility only up to the square root is efficient because if a number \( n \) has a factor larger than \( \sqrt{n} \), it must also have a factor smaller than \( \sqrt{n} \).

Determine whether the integer 179 is prime or composite using an approximation of its square root (w ≈ 13.4).

Learn how to determine the primality of the integer 179 by using the square root approximation method. This problem covers prime number properties, divisibility rules, and provides a step-by-step solution for checking prime numbers up to the square root approximation (13.4). Suitable for high school students.

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